Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a0 + a1x + ... + akxk is a k-degree polynomial with integral coefficients such that (p, a0, a1, ..., ak) = 1, for any integer m, we study the asymptotic property of $$\sum\limits_{\chi \ne \chi _0 } {\left| {\sum\limits_{a = 1}^{p - 1} {\chi (a)e\left( {\frac{{f(a)}}{p}} \right)} } \right|^2 \left| {L(1,\chi )} \right|^{2m} } ,$$ where e(y) = e2πiy. The main purpose is to use the analytic method to study the 2m-th power mean of Dirichlet L-functions with the weight of the general trigonometric sums and give an interesting asymptotic formula. This result is an extension of the previous results.